Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rossspw Structured version   Visualization version   GIF version

Theorem rossspw 31707
Description: A ring of sets is a collection of subsets of 𝑂. (Contributed by Thierry Arnoux, 18-Jul-2020.)
Hypothesis
Ref Expression
isros.1 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}
Assertion
Ref Expression
rossspw (𝑆𝑄𝑆 ⊆ 𝒫 𝑂)
Distinct variable groups:   𝑂,𝑠   𝑆,𝑠,𝑥,𝑦
Allowed substitution hints:   𝑄(𝑥,𝑦,𝑠)   𝑂(𝑥,𝑦)

Proof of Theorem rossspw
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isros.1 . . . 4 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}
21isros 31706 . . 3 (𝑆𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆)))
32simp1bi 1146 . 2 (𝑆𝑄𝑆 ∈ 𝒫 𝒫 𝑂)
43elpwid 4499 1 (𝑆𝑄𝑆 ⊆ 𝒫 𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  wral 3053  {crab 3057  cdif 3840  cun 3841  wss 3843  c0 4211  𝒫 cpw 4488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pw 4490
This theorem is referenced by:  rossros  31718
  Copyright terms: Public domain W3C validator